In category theory the Chu construction is a method of generating a \(*\) -autonomous category from a closed symmetric
monoidal category (CSMC) (see [1]} and [2]}). The \(*\) -autonomous categories provide models for classical (multiplicative) linear logic (in [3]}). The Chu construction was introduced by Chu in his Master's Thesis,
and appeared first in [4]}. The category \(\textnormal {\textbf {Chu}}(\mathrm {\mathbf {Set}}, X)\)  was introduced by Lafont and Streicher in [5]} under the name of games (semantics for linear logic). In a series of papers, see e.g., [6]}, Pratt and his collaborators applied the Chu construction on topics of theoretical computer science (e.g., concurrency). The Chu construction has been applied to hardware verification, game theory, fuzzy systems, and the foundations of quantum mechanics (see [7]} and [8]}).
There is a rich representation theory connected to the Chu construction, as many important, and quite different categories are
represented (fully embedded) into some Chu category. The representation of categories related to constructive mathematics into some Chu category is a major theme of this paper.
